7.3.2.

Do two processes have the same standard deviation?

Testing hypotheses related to standard deviations from two
processes

Given two random samples of measurements,
$$ Y_1, \, \ldots, \, Y_N \,\,\,\,\, \mbox{ and } \,\,\,\,\,
Z_1, \, \ldots, \, Z_N $$
from two independent processes, there are three types of questions
regarding the true standard deviations of the processes that can be
addressed with the sample data. They are:

Are the standard deviations from the two processes the same?

Is the standard deviation of one process less than the
standard deviation of the other process?

Is the standard deviation of one process greater than the
standard deviation of the other process?

The \(F\)
distribution has the property that
$$ F_{1-\alpha/2, \, \nu_1, \, \nu_2}
= \frac{1}{F_{\alpha/2, \, \nu_2, \, \nu_1}} \, ,$$
which means that only upper critical values are required for two-sided
tests. However, note that the degrees of freedom are interchanged in
the ratio. For example, for a two-sided test at significance level
0.05, go to the \(F\)
table labeled "2.5 % significance level".

For \(F_{\alpha/2, \, \nu_2, \, \nu_1}\),
reverse the order of the degrees of freedom; i.e.,
look across the top of the table for \(\nu_2 = N_2-1\)
and down the table for \(\nu_1 = N_1-1\)

For \(F_{\alpha/2, \, \nu_1, \, \nu_2}\),
look across the top of the table for \(\nu_2 = N_2-1\)
and down the table for \(\nu_1 = N_1-1\)

Critical values for cases (2) and (3) are defined similarly, except
that the critical values for the one-sided tests are based on \(\alpha\)
rather than on \(\alpha/2\).

Two-sided confidence interval

The two-sided confidence interval for the ratio of the two unknown
variances (squares of the standard deviations) is shown below.

One interpretation of the confidence interval is that if the quantity
"one" is contained within the interval, the standard deviations are
equivalent.

Example of unequal number of data points

A new procedure to assemble a device is introduced and tested for
possible improvement in time of assembly. The question being
addressed is whether the standard deviation, \(\sigma_2\),
of the new
assembly process is better (i.e., smaller) than the standard deviation,
\(\sigma_1\),
for the old assembly process. Therefore, we test the null hypothesis that
\(\sigma_1 \le \sigma_2\).
We form the hypothesis in this way because we hope to reject it,
and therefore accept the alternative that \(\sigma_2\)
is less than \(\sigma_1\).
This is hypothesis (2). Data
(in minutes required to assemble a
device) for both the old and new processes are listed on an
earlier page. Relevant statistics are shown below: